Involutive distribution

The geometric interpretation of a completely-integrable differential system on an $ n $-dimensional differentiable manifold $ M ^ {n} $ of class $ C ^ {k} $, $ k \geq 3 $. A $ p $-dimensional distribution (or a differential system of dimension $ p $) of class $ C ^ {r} $, $ 1 \leq r < k $, on $ M ^ {n} $ is a function associating to each point $ x \in M ^ {n} $ a $ p $-dimensional linear subspace $ D( x) $ of the tangent space $ T _ {x} ( M ^ {n} ) $ such that $ x $ has a neighbourhood $ U $ with $ p $ $ C ^ {r} $ vector fields $ X _ {1}, \dots, X _ {p} $ on it for which the vectors $ X _ {1} ( y), \dots, X _ {p} ( y) $ form a basis of the space $ D ( y) $ at each point $ y \in U $. The distribution $ D $ is said to be involutive if for all points $ y \in U $,

$$ [ X _ {i} , X _ {j} ] ( y) \in D ( y) ,\ \ 1 \leq i , j \leq p . $$

This condition can also be stated in terms of differential forms. The distribution $ D $ is characterized by the fact that

$$ D ( y) = \{ {X \in T _ {y} ( M ^ {n} ) } : { \omega ^ \alpha ( y) ( X) = 0 } \} ,\ p < \alpha \leq n , $$

where $ \omega ^ {p+1}, \dots, \omega ^ {n} $ are $ 1 $-forms of class $ C ^ {r} $, linearly independent at each point $ x \in U $; in other words, $ D $ is locally equivalent to the system of differential equations $ \omega ^ \alpha = 0 $. Then $ D $ is an involutive distribution if there exist $ 1 $-forms $ \omega _ \beta ^ \alpha $ on $ U $ such that

$$ d \omega ^ \alpha = \ \sum _ {\beta = p + 1 } ^ { n } \omega ^ \beta \wedge \omega _ \beta ^ \alpha , $$

that is, the exterior differentials $ d \omega ^ \alpha $ belong to the ideal generated by the forms $ \omega ^ \beta $.

A distribution $ D $ of class $ C ^ {r} $ on $ M ^ {n} $ is involutive if and only if (as a differential system) it is an integrable system (Frobenius' theorem).

References

[1]	C. Chevalley, "Theory of Lie groups" , 1 , Princeton Univ. Press (1946)
[2]	R. Narasimhan, "Analysis on real and complex manifolds" , North-Holland & Masson (1968) (Translated from French)